Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't get any response so hoping I could get some sort of help here.

Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime.

Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$).

Then Artin-Tate claims that $\frac{dt}{dt_P}$ is almost always a local unit. ie for almost all $P$ $v_P(\frac{dt}{dt_P})=0$. The book refers to chapter 17-4 of Algebraic numbers and algebraic functions but I really couldn't find any relevance.

It is clear that $v_P(t)=0$ for almost all $t$ so I just need to verify that for each expansion of $t$ at $k_P$, say

$$t=\sum_{i\geq0}a_i t_P^{i},$$ $a_1$ is almost always non-zero.

But I have absolutely no clue how to prove it. Or I am also thinking the results in chapter 17-4 of Artin's book could actually be useful but I don't really see it.

Any help/reference would be great!

It turns out just by simple calculation this is easy to prove if $K$ is a rational function field. However it still puzzles me how to prove it if $K$ is a general function field. I am sure this fact is supposed to be classically known for a long time but any standard book which discusses global function field does not seem to discuss this.

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (https://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't get any response so hoping I could get some sort of help here.

Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime.

Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$).

Then Artin-Tate claims that $\frac{dt}{dt_P}$ is almost always a local unit. ie for almost all $P$ $v_P(\frac{dt}{dt_P})=0$. The book refers to chapter 17-4 of Algebraic numbers and algebraic functions but I really couldn't find any relevance.

It is clear that $v_P(t)=0$ for almost all $t$ so I just need to verify that for each expansion of $t$ at $k_P$, say

$$t=\sum_{i\geq0}a_i t_P^{i},$$ $a_1$ is almost always non-zero.

But I have absolutely no clue how to prove it. Or I am also thinking the results in chapter 17-4 of Artin's book could actually be useful but I don't really see it.

Any help/reference would be great!

It turns out just by simple calculation this is easy to prove if $K$ is a rational function field. However it still puzzles me how to prove it if $K$ is a general function field. I am sure this fact is supposed to be classically known for a long time but any standard book which discusses global function field does not seem to discuss this.

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't get any response so hoping I could get some sort of help here.

Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime.

Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$).

Then Artin-Tate claims that $\frac{dt}{dt_P}$ is almost always a local unit. ie for almost all $P$ $v_P(\frac{dt}{dt_P})=0$. The book refers to chapter 17-4 of Algebraic numbers and algebraic functions but I really couldn't find any relevance.

It is clear that $v_P(t)=0$ for almost all $t$ so I just need to verify that for each expansion of $t$ at $k_P$, say

$$t=\sum_{i\geq0}a_i t_P^{i},$$ $a_1$ is almost always non-zero.

But I have absolutely no clue how to prove it. Or I am also thinking the results in chapter 17-4 of Artin's book could actually be useful but I don't really see it.

Any help/reference would be great!

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (http://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn't get any response so hoping I could get some sort of help here.

Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime.

Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$).

Then Artin-Tate claims that $\frac{dt}{dt_P}$ is almost always a local unit. ie for almost all $P$ $v_P(\frac{dt}{dt_P})=0$. The book refers to chapter 17-4 of Algebraic numbers and algebraic functions but I really couldn't find any relevance.

It is clear that $v_P(t)=0$ for almost all $t$ so I just need to verify that for each expansion of $t$ at $k_P$, say

$$t=\sum_{i\geq0}a_i t_P^{i},$$ $a_1$ is almost always non-zero.

But I have absolutely no clue how to prove it. Or I am also thinking the results in chapter 17-4 of Artin's book could actually be useful but I don't really see it.

Any help/reference would be great!

It turns out just by simple calculation this is easy to prove if $K$ is a rational function field. However it still puzzles me how to prove it if $K$ is a general function field. I am sure this fact is supposed to be classically known for a long time but any standard book which discusses global function field does not seem to discuss this.